# How does the order between limsup and summation affect the result?

Given a double-indexed real sequence $\{ x_{n,m}\}$, do we have

$$\limsup_{n} \sum_m x_{n,m} \leq \sum_m \limsup_{n} \, x_{n,m}$$

$$\liminf_{n} \sum_m x_{n,m} \geq \sum_m \liminf_{n} \,x_{n,m}?$$

I am not sure about these, and just have some guess based on how sup and sum commute.

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If the xs are nonnegative, the sum of the liminfs is at most the liminf of the sums. This is Fatou's lemma, see here: en.wikipedia.org/wiki/Fatou%27s_lemma which explains some weaker hypothesis and the corresponding statement for limsups. –  Did Aug 25 '11 at 14:51
What do we do if the series $\sum_{m=0}^{\infty}x_{n,m}$ is not convergent? (if the $x_{n,m}$ are not necessary nonnegative, the sequence $\{\sum_{m=0}^Nx_{n,m}\}$ may have no limit at all, even in $\overline{\mathbb R}$). –  Davide Giraudo Aug 25 '11 at 15:15